F凸函数的Hermite-Hadamard型不等式

Hermite-Hadamard Type Inequalities for F-Convex Functions

首先证明:若F在广义凸集K上满足条件P_1,P_2,f:K→R在K上是F凸函数,则对任意b,a∈K,f[F(b,a,λ)]关于λ是[0,1]上的凸函数。然后利用条件P_1,P_2,F凸函数的定义以及凸函数的Jensen不等式,建立了与f[F(b,a,λ)]关于严格单调增加函数的黎曼积分有关的不等式,从而获得F凸函数的新的Hermite-Hadamard型不等式。由此在特殊情况下得到凸函数、预不变凸函数、GA凸函数的加权Hermite-Hadamard型不等式。

Firstly, it is pointed out that if F satisfies conditions Pl ,P2 over generalized convex set K, f： K→R is F-convex function on K, then .f[F（x, y,λ） ] is convex function over λ on [0, 1] for any x, y ∈ K. Secondly, an inequality associated with Riemann integral of [F（x,y,λ）] over strictly monotone increasing functions is established by using conditions P1 ,P2, the definition and Jensen＇s inequality of convex functions, therefore, new Hermite-Hadamard type inequalities for F-convex functions are obtained. Finally, under special conditions, weighted Hermite-Hadamard type inequalities for convex functions, preinvex functions, GA-convex func tions are derived.